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We investigate an M-channel critically sampled filter bank for graph signals where each of the M filters is supported on a different subband of the graph Laplacian spectrum. We partition the graph vertices such that the mth set comprises a uniqueness set for signals supported on the mth subband. For analysis, the graph signal is filtered on each subband and downsampled on the corresponding set of vertices.

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Historically, sparse methods and neural networks, particularly modern deep learning methods, have been relatively disparate areas. Sparse methods are typically used for signal enhancement, compression,and recovery, usually in an unsupervised framework, while neural networks commonly rely on a supervised training set.

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The success of Empirical Mode Decomposition (EMD) resides in its practical approach to dissect non-stationary data. EMD repetitively goes through the entire data span to iteratively extract Intrinsic Mode Functions (IMFs). This approach, however, is not suitable for data stream as the entire data set has to be reconsidered every time a new point is added. To overcome this, we propose Online EMD, an algorithm that extracts IMFs on the fly.

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Graph Signal Processing generalizes classical signal processing to signal or data indexed by the vertices of a weighted graph. So far, the research efforts have been focused on static graph signals. However numerous applications involve graph signals evolving in time, such as spreading or propagation of waves on a network. The analysis of this type of data requires a new set of methods that takes into account the time and graph dimensions. We propose a novel class of wavelet frames named Dynamic Graph Wavelets, whose time-vertex evolution follows a dynamic process.

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Sampling and reconstruction of bandlimited graph signals have well-appreciated merits for dimensionality reduction, affordable storage, and online processing of streaming network data. However, these parsimonious signals are oftentimes encountered with high-dimensional linear inverse problems. Hence, interest shifts from reconstructing the signal itself towards instead approximating the input to a prescribed linear operator efficiently.

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